First-order formalism of holographic Wilsonian renormalization group: Langevin equation

We study a mathematical relationship between holographic Wilsonian renormalization group and stochastic quantization framework. We extend the original proposal given in arXiv:1209.2242 to interacting theories. The original proposal suggests that fictitious time (or stochastic time) evolution of stochastic 2-point correlation function will be identical to the radial evolution of the double-trace operator of certain classes of holographic models, which are free theories in AdS space. We study holographic gravity models with interactions in AdS space, and establish a map between the holographic renormalization flow of multi-trace operators and stochastic n-point functions. To give precise examples, we extensively study conformally coupled scalar theory in AdS6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {AdS}_6$$\end{document}. What we have found is that the stochastic time t dependent 3-point function obtained from Langevin equation with its Euclidean action being given by SE=2Ios\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_E=2I_{os}$$\end{document} is identical to holographic renormalization group evolution of holographic triple-trace operator as its energy scale r changes once an identification of t=r\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t=r$$\end{document} is made. Ios\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_{os}$$\end{document} is the on-shell action of holographic model of conformally coupled scalar theory at the AdS boundary. We argue that this can be fully extended to mathematical relationship between multi-point functions and multi-trace operators in each framework.